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Smoothed Analysis of the k-Swap Neighborhood for Makespan Scheduling

Lars Rohwedder, Ashkan Safari, Tjark Vredeveld

TL;DR

The smoothed number of iterations required to find a local optimum with respect to the k-swap neighborhood is bounded by O(m^2 \cdot n^{2k+2} \cdot \log m \cdot \phi)$, where $n$ and $m$ are the numbers of jobs and machines, respectively, and $\phi \geq 1$ is the perturbation parameter.

Abstract

Local search is a widely used technique for tackling challenging optimization problems, offering simplicity and strong empirical performance across various problem domains. In this paper, we address the problem of scheduling a set of jobs on identical parallel machines with the objective of makespan minimization, by considering a local search neighborhood, called $k$-swap. A $k$-swap neighbor is obtained by interchanging the machine allocations of at most $k$ jobs scheduled on two machines. While local search algorithms often perform well in practice, they can exhibit poor worst-case performance. In our previous study, we showed that for $k \geq 3$, there exists an instance where the number of iterations required to converge to a local optimum is exponential in the number of jobs. Motivated by this discrepancy between theoretical worst-case bound and practical performance, we apply smoothed analysis to the $k$-swap local search. Smoothed analysis has emerged as a powerful framework for analyzing the behavior of algorithms, aiming to bridge the gap between poor worst-case and good empirical performance. In this paper, we show that the smoothed number of iterations required to find a local optimum with respect to the $k$-swap neighborhood is bounded by $O(m^2 \cdot n^{2k+2} \cdot \log m \cdot φ)$, where $n$ and $m$ are the numbers of jobs and machines, respectively, and $φ\geq 1$ is the perturbation parameter. The bound on the smoothed number of iterations demonstrates that the proposed lower bound reflects a pessimistic scenario which is rare in practice.

Smoothed Analysis of the k-Swap Neighborhood for Makespan Scheduling

TL;DR

The smoothed number of iterations required to find a local optimum with respect to the k-swap neighborhood is bounded by O(m^2 \cdot n^{2k+2} \cdot \log m \cdot \phi)nm\phi \geq 1$ is the perturbation parameter.

Abstract

Local search is a widely used technique for tackling challenging optimization problems, offering simplicity and strong empirical performance across various problem domains. In this paper, we address the problem of scheduling a set of jobs on identical parallel machines with the objective of makespan minimization, by considering a local search neighborhood, called -swap. A -swap neighbor is obtained by interchanging the machine allocations of at most jobs scheduled on two machines. While local search algorithms often perform well in practice, they can exhibit poor worst-case performance. In our previous study, we showed that for , there exists an instance where the number of iterations required to converge to a local optimum is exponential in the number of jobs. Motivated by this discrepancy between theoretical worst-case bound and practical performance, we apply smoothed analysis to the -swap local search. Smoothed analysis has emerged as a powerful framework for analyzing the behavior of algorithms, aiming to bridge the gap between poor worst-case and good empirical performance. In this paper, we show that the smoothed number of iterations required to find a local optimum with respect to the -swap neighborhood is bounded by , where and are the numbers of jobs and machines, respectively, and is the perturbation parameter. The bound on the smoothed number of iterations demonstrates that the proposed lower bound reflects a pessimistic scenario which is rare in practice.

Paper Structure

This paper contains 3 sections, 9 theorems, 8 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and Notation
  3. Smoothed Analysis

Key Result

Lemma 3.1

For all the $\ell$min-load machines in the set $\gamma_s$, we have $L_{\ell\min} (t+1) \geq L_{\ell\min} (t)$ when a $k$-swap operation of type-1 occurs in iteration $t$.

Figures (2)

  • Figure 1: Jump (left diagram) and swap (right diagram) operators.
  • Figure 2: (a) Type-1 of improving $k$-swap. After swapping $A$ with $B$, $i'$ moves to $\gamma_l$. (b) Type-2 of improving $k$-swap. After swapping $A$ with $B$, $i'$ stays in $\gamma_s$. Note that in both of the cases, machine $i$ does not necessarily move to $\gamma_s$ after the swap.

Theorems & Definitions (15)

  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Corollary 3.3
  • Lemma 3.5
  • proof
  • Lemma 3.7
  • proof
  • Corollary 3.8
  • ...and 5 more